2360 ieee transactions on wireless communications, …

2360 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 4, APRIL 2018

Index Modulated OFDM Spread SpectrumQiang Li , Miaowen Wen , Member, IEEE, Ertugrul Basar , Senior Member, IEEE,

and Fangjiong Chen, Member, IEEE

Abstract— In this paper, we propose an index modulatedorthogonal frequency division multiplexing spread spectrum(IM-OFDM-SS) scheme, which combines the techniques of SSand IM under the framework of OFDM. In this scheme,the information bits are jointly conveyed by the indices ofspreading codes and the conventional M-ary modulated symbols.A low-complexity maximal ratio combining (MRC) detector isdesigned, in which the receiver first detects the spreading codesand then de-spreads and demodulates the symbols. The biterror rate (BER) performance of IM-OFDM-SS systems in thepresence of channel estimation errors is analyzed. An upperbound and approximate average bit error probability associatedwith maximum-likelihood and MRC detection, respectively, arederived. Subsequently, we extend the idea of IM-OFDM-SSto multi-code and multi-user scenarios, proposing generalized(G-)IM-OFDM-SS and IM-based multi-carrier code divi-sion multiple access (IM-MC-CDMA), respectively. Simulationresults verify the analyses and show that the proposed IM/GIM-OFDM-SS outperforms the existing OFDM-IM and OFDM-SS schemes significantly. In addition, the IM-MC-CDMA exhibitsa lower BER than MC-CDMA at the same SEs for eithermulti-user or single-user detection.

Index Terms— OFDM, index modulation, spread spectrum,channel estimation errors, multi-user communications.

I. INTRODUCTION

W ITH the increasing demand for high data rate commu-nications, multi-carrier transmission has received wide

interest. The first orthogonal frequency division multiplex-ing (OFDM) scheme was proposed in 1966 [1], which presents

Manuscript received June 9, 2017; revised September 20, 2017 andNovember 15, 2017; accepted January 9, 2018. Date of publication January 23,2018; date of current version April 8, 2018. This work was supported in partby the National Natural Science Foundation of China under Grant 61671211,Grant U1701265, and Grant 61501190, in part by the Natural ScienceFoundation of Guangdong Province under Grant 2016A030311024, in partby the State Major Science and Technology Special Projects under Grant2016ZX03001009 and Grant 2017ZX03001001, in part by the open researchfund of the National Mobile Communications Research Laboratory, SoutheastUniversity, under Grant 2017D08, in part by the Turkish Academy of SciencesOutstanding Young Scientist Award Programme (TUBA-GEBIP), and in partby the Scientific Research Projects Foundation, Istanbul Technical University,under Grant 40607. The associate editor coordinating the review of this paperand approving it for publication was M. R. Nakhai. (Corresponding author:Miaowen Wen.)

Q. Li and F. Chen are with the School of Electronic and InformationEngineering, South China University of Technology, Guangzhou 510640,China (e-mail: [email protected]; [email protected]).

M. Wen is with the School of Electronic and Information Engineering,South China University of Technology, Guangzhou 510640, China, and alsowith the National Mobile Communications Research Laboratory, SoutheastUniversity, Nanjing 210096, China (e-mail: [email protected]).

E. Basar is with the Faculty of Electrical and Electronics Engineering,Istanbul Technical University, 34469 Istanbul, Turkey (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TWC.2018.2793238

the principle of orthogonal multiplexing for transmitting anumber of information bits simultaneously without interchan-nel and intersymbol interferences. In [2], discrete Fouriertransform (DFT) is used for baseband processing rather thana bank of oscillators, such that the implementation com-plexity of OFDM systems is greatly reduced. Furthermore,OFDM achieves a high spectral efficiency (SE) due to thedensely spaced subcarriers with overlapping spectrum andthe avoidance of guard bands. Hence, OFDM is adoptedin many telecommunication standards such as digital videobroadcasting standard for terrestrial broadcasting [3], LongTerm Evolution (LTE) and IEEE 802.11x [4].

Recently, the concept of index modulation (IM), whichutilizes the index(es) of some transmission entities to carryextra information bits, has been proposed as a competitivealternative digital modulation technique for the future wirelesscommunications [5]. Inspired by the spatial modulation (SM),which considers an M-ary modulated symbol and the indexof an active antenna as the information-bearing units [6],a number of researchers also applied the IM principle toOFDM systems. In subcarrier-index modulation (SIM-)OFDMscheme [7], one index bit is offered to each subcarrier and thesubcarriers with the majority bit-value are activated to transmitsymbols. The number of active subcarriers in each OFDMblock is variable and a perfect feedforward link is neededto signify the majority bit-value to the receiver, which makethis scheme less practical. Enhanced (E-)SIM-OFDM avoidsthese drawbacks by employing one index bit to control thestates of two consecutive subcarriers [8]. However, the SE ofESIM-OFDM systems is far behind that of OFDM systemsif the same constellation size is adopted. A more flexibleand efficient IM scheme for OFDM named OFDM withIM (OFDM-IM) is proposed in [9], in which the total subcar-riers are divided into several blocks and a subset of subcarrierswithin each block are activated according to the index bits totransmit M-ary modulated symbols.

The performance of OFDM-IM in terms of bit errorrate (BER) [9], minimum Euclidean distance [10], achievablerate [11], and inter-carrier interference [12] is analyzed, reveal-ing the superiority of OFDM-IM over its OFDM counterpart.Due to the advantages of OFDM-IM, plenty of researchon OFDM-IM and its variants have emerged. Apart fromthe maximum-likelihood (ML) detector, some low-complexitydetectors, such as log-likelihood ratio detector [9] and greedydetector [13], are proposed to ease the computational burdenon the receiver. An upper bound and an approximate expres-sion for the bit error probability (BEP) of OFDM-IM arederived in [14] and [15] assuming ML and energy detection,respectively.

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LI et al.: INDEX MODULATED OFDM SPREAD SPECTRUM 2361

Based on the framework of OFDM-IM, some effortsto increase its SE and diversity order have been madein the literature. On the issue of SE, two generalizationsof OFDM-IM, termed OFDM with generalized indexmodulation 1 (OFDM-GIM1) and OFDM-GIM2, are proposedin [16]. In OFDM-GIM1, the number of active subcarriersin each block is changeable per transmission accordingto the index bits. In OFDM-GIM2, the IM technique isperformed independently on the in-phase (I-) and quadrature(Q-) dimensions, which doubles the amount of index bitsof OFDM-IM. Compressed sensing (CS) assisted OFDM-IM [17] improves the SE of OFDM-IM by implementingthe conventional IM in a high-dimensional virtual domain,and then compressing the resulting IM symbols into thelow-dimensional subcarriers in the frequency domain with thehelp of the CS technique. In multiple-input multiple-output(MIMO-)OFDM-IM scheme [18], each transmit antenna isequipped with an OFDM-IM transmitter such that the SEincreases with the number of transmit antennas. A low-complexity yet near-optimal detector for MIMO-OFDM-IM isdesigned in [19] based on the sequential Monte Carlo theory.A generic IM scheme for MIMO-OFDM systems, calledgeneralized space and frequency index modulation (GSFIM) isproposed in [20], performing IM in both spatial and frequencydomains, and its low-complexity detector is provided in [21]using the multi-stage message passing approach. In dual-modeIM-aided OFDM (DM-OFDM) [22], the “inactive” subcarriersin conventional OFDM-IM are re-activated intentionally totransmit symbols drawn from a constellation different fromthat employed by the “active” subcarriers.

On the other hand, there already exist some enhancedschemes improving the asymptotic BER performance ofOFDM-IM and related systems. In [23], the subcarrier-level interleaving is introduced into OFDM-IM to obtainfrequency diversity. In coordinate interleaved (CI-)OFDM-IM [24], the real and imaginary parts of each complex datasymbol are transmitted on two different active subcarriersusing the CI orthogonal design, which increases the transmitdiversity order of ordinary modulation bits from unity to two.To increase the transmit diversity order of index bits solely,the same index bits are fed into two or more blocks in [25].Space-frequency code is constructed in [26] to improve thetransmit diversity of MIMO-OFDM-IM. Finally, the symbolerror probability of OFDM-IM with diversity reception isanalyzed in [27], in which greedy detection is employed forindex bits and ML detection for ordinary modulation bits.

Spread spectrum (SS) signals are widely used forthe transmission of digital information over some radiochannels to offer message privacy, the capability ofcombating the determined interference, and code divisionmultiple access (CDMA). The combination of multi-carriermodulation and SS, termed multi-carrier (MC-)SS, is anattempt to take advantage of attributes of both techniques [28].This combination has prompted a number of new multipleaccess schemes, such as MC-CDMA and MC direct sequenceCDMA (MC-DS-CDMA) [29], [30]. In MC-CDMA, the chipsof a spread data symbol are mapped in the frequency scaleover several parallel subcarriers. By contrast, MC-DS-CDMA

converts the high-rate data stream into parallel low-ratesub-streams, and then the DS spreading is performed on eachsub-channel in the time scale with a user-specific spreadingcode. As a special form of MC-DS-CDMA, single-carrierfrequency division multiple access (SC-FDMA) has beenadopted in LTE-Advanced for the uplink transmission [31].In SC-FDMA, the time-domain data symbols are transformedto the frequency domain by a DFT precoder before carryingout the MC modulation. In the above-mentioned MC-CDMAand MC-DS-CDMA, the spreading codes are used to spreadsymbols and they do not carry any information.

The idea of code IM (CIM), which utilizes the indicesof spreading codes to convey additional information bits,is first proposed by the authors of [32]. In their proposedscheme, one additional bit on the I- (Q-) branch is usedto select one of two spreading codes, by which the I- (Q-)component of a quadrature phase shift keying (QPSK) symbolis spread. Later, the authors from the same research groupgeneralize CIM in [33] by employing M-PSK/quadratureamplitude modulation (QAM) and allowing the number ofavailable spreading codes to be an arbitrary integral powerof two. Furthermore, [34] validates the superiority of CIMover SM in terms of energy efficiency, system complexity, andBER performance, while [35] introduces the CIM techniqueinto differential chaos shift keying scheme to obtain the betterBER performance. However, in the existing CIM schemes, DSspreading is performed in the time scale for single-carrier andsingle-user communications.

Different from the previous CIM schemes, a novel CIMscheme based on OFDM systems, termed index modu-lated OFDM-SS (IM-OFDM-SS), is proposed in this paper,to improve the diversity order of OFDM-IM and the SE ofOFDM-SS. Moreover, a framework of CIM-based multi-usercommunications is provided. The contributions of this paperare summarized as follows:

• The techniques of IM and SS are combined under theframework of OFDM for the first time. In IM-OFDM-SS, a data symbol is spread across several subcarriersto harvest additional diversity gain by the spreadingcode that is selected from a predefined set according tothe index bits. The idea of IM-OFDM-SS is extendedto the multi-code and multi-user scenarios, resulting inthe generalized (G-)IM-OFDM-SS and IM-MC-CDMAschemes, respectively.

• We propose a low-complexity maximal ratiocombining (MRC) detector for IM-OFDM-SS, whichexhibits near-ML error performance. Taking into accountthe channel estimation errors, an upper bound and anapproximate mathematical expression for the BEP ofIM-OFDM-SS are derived, assuming ML and MRCdetection, respectively.

• Extensive computer simulations are performed, whoseresults show that IM/GIM-OFDM-SS outperformsthe existing OFDM, OFDM-IM, and OFDM-SSschemes significantly. In multi-user communications,IM-MC-CDMA also performs better than theconventional MC-CDMA at the same SEs for eithermulti-user or single-user detection.


Fig. 1. Transmitter structure of IM-OFDM-SS.

The rest of this paper is organized as follows. Section IIgives the system model of IM-OFDM-SS, including the trans-mitter and receiver structures. The performance in terms ofupper bound and mathematical expression for BEP is analyzedin Section III. The idea of IM-OFDM-SS is extended to multi-code and multi-user scenarios in Section IV. Section V presentsthe computer simulation results and detection complexitycomparisons, followed by the conclusion in Section VI.

Notation: Column vectors and matrices are denoted by low-ercase and capital bold letters, respectively. Superscripts ∗, T ,and H stand for conjugate, transpose, and Hermitian transpose,respectively. C denotes the ring of complex numbers. det(·)and rank(·) return the determinant and rank of a matrix,respectively. diag(·) transforms a vector into a diagonal matrix.CN (0, σ 2) represents the complex Gaussian distribution withzero mean and variance σ 2. The probability of an eventand the probability density function (PDF) are denoted byPr(·) and f (·), respectively. E{·} denotes the expectation oper-ation. Q(·), Q1(·, ·), and I0(·) are the Gaussian Q-function,the first-order Marcum Q-function, and the zero-order modi-fied Bessel function of first kind, respectively. In is the n × nidentity matrix. �·�, C(·, ·), and ‖·‖ denote the floor function,the binomial coefficient, and the Frobenius norm, respectively.We define [x1, . . . , xn]T ⊗[y1, . . . , yn]T = [x1y1, . . . , xn yn]T .

II. SYSTEM MODEL

In this section, we present the transmitter and receiverstructures of IM-OFDM-SS.

A. Transmitter

The transmitter structure of IM-OFDM-SS based on theframework of OFDM with N subcarriers is depicted in Fig. 1.To perform IM efficiently, m information bits to be sent foreach OFDM frame duration are equally separated into g blocksand IM is performed within n = N/g subcarriers in a block-wise manner. Since the process in all blocks are the same andindependent of each other, let us take the β-th block as anexample, where β ∈ {1, . . . , g}. In the β-th block, the availablep = m/g bits are further divided into two parts. The first partconsisting of p1 index bits are used to select the spreadingcode ci(β) ∈ Cn×1 from a predefined set C = {c1, . . . , cn},

where i (β) ∈ {1, . . . , n} is the index of the spreading code forthe β-th block. The second part with p2 = log2(M) symbolbits are mapped into a symbol s(β) ∈ X , where X is an M-arypulse amplitude modulation (PAM)/QAM constellation havingunit average power.

To enable low-complexity detection at the receiver,the spreading codes in C should be mutually orthogonal,which limits the maximum size of C to be n. In thispaper, Zadoff-Chu and Walsh orthogonal codes are con-sidered. For Walsh codes, we have C = {C:,1, . . . , C:,n},where C:,k, k ∈ {1, . . . , n}, represents the k-th columnof the n × n Hadamard matrix. For Zadoff-Chu codes,C is derived by cyclically shifting a basic Zadoff-Chusequence by 0, . . . , n − 1 positions [36]. For n beingan even number, we employ the Zadoff-Chu sequence,{1, exp( j �π

n ), exp( j 4�πn ), . . . , exp( j �π(n−1)2

n )}, where � isan integer relatively prime to n. Due to the length constraintof Walsh codes, n is assumed to be an integer power of 2.Therefore, we set p1 = log2(n) and the mapping and de-mapping between p1 bits and i (β) can be easily implementedby the natural binary code [37].

Then, the symbol s(β) is spread across n subcarriers by theselected spreading code ci(β) , yielding

x(β) =[x (β)

1 , . . . , x (β)n

]T = s(β)ci(β)

=[s(β)ci(β),1, . . . , s(β)ci(β),n

]T, (1)

where ci(β),k, k ∈ {1, . . . , n} is the k-th element of ci(β) . Afterobtaining x(β) for all β, the OFDM block creator concatenatesthem, yielding the N × 1 main OFDM block as follows:

xF = [x1, . . . , xN ]T

=[x (1)

1 , . . . , x (1)n , x (2)

1 , . . . , x (2)n , . . . , x (g)

1 , . . . , x (g)n

]T.

(2)

To harvest the frequency diversity, an interleaver is adopted,which makes the symbols in the x(β) spaced equally by gsubcarriers, obtaining

x′F = [

x ′1, . . . , x ′

N

]T

=[x (1)

1 , . . . , x (g)1 , x (1)

2 , . . . , x (g)2 , . . . , x (1)

n , . . . , x (g)n

]T.

(3)

Afterwards, the remaining procedures are the same as thoseof classical OFDM, which are omitted in Fig. 1. First, x′

Fis processed by the inverse fast Fourier transform (IFFT),yielding the time-domain OFDM block

xT = [X1, . . . , X N ]T = 1√N

WHN x′

F , (4)

where WN is the N × N DFT matrix with WHN WN = NIN .

A length-L cyclic prefix (CP) [X N−L+1, . . . , X N ]T is added tothe beginning of xT . After the parallel-to-serial and digital-to-analog conversions, the data is transmitted over the frequency-selective Rayleigh fading channel, whose impulse response isgiven by hT = [

hT ,1, . . . , hT ,v

]T , where v is the number ofchannel taps and each entry of hT is a circularly symmetric


complex Gaussian random variable following the distributionCN (0, 1/v). Note that L is chosen to be larger than v tocombat the intersymbol interference.

At the receiver, N-point FFT and de-interleaving are carriedout in sequence after discarding the CP of the received signal.Hence, the input-output relationship in the frequency domainfor the β-th block is given by

y(β) = X(β)h(β) + w(β), (5)

where X(β) = diag(x(β)), h(β) is the n × 1 frequency-domainchannel vector, and w(β) ∈ Cn×1 is the noise vector inthe frequency domain with the distribution CN (0, N0In).Note that the frequency-domain channel vector is given byhF = WN h0

T , where h0T is the length-N zero-padded version

of the vector hT , namely h0T = [hT

T , 0, . . . , 0]T . Hence, withde-interleaving, h(β) = [hF,β , hF,β+g , . . . , hF,β+(n−1)g]T ,and the elements of h(β) can be viewed as independent andidentically distributed (i.i.d.), which means h(β) ∼ CN (0, In).We rewrite (5) in a scalar form as

y(β)k = s(β)h(β)

k ci(β),k + w(β)k , k = 1, . . . , n, (6)

where y(β)k , h(β)

k , and w(β)k are the samples of y(β), h(β),

and w(β), respectively, at the k-th subcarrier. We definethe average signal-to-noise ratio (SNR) per subcarrier asγ = 1/N0. The SE of IM-OFDM-SS scheme regardless ofthe CP overhead is given by (bps/Hz)

EIM−OFDM−SS = (p1 + p2)/n = log2 (Mn) /n, (7)

while the SE of OFDM-SS is EOFDM−SS = log2 (M) /n, fromwhich we observe that IM-OFDM-SS achieves a higher SEthan OFDM-SS.

B. Receiver

In this subsection, we address the detection problem ofIM-OFDM-SS by taking into account the effects of channelestimation errors. Two different types of detectors are providedfor the IM-OFDM-SS scheme, including the ML detector andthe MRC detector, whose theoretical error performance willbe analyzed in Section III. Since the encoding processes forall blocks are identical and independent, the detection can beperformed block by block. To simplify notations, in the sequelwe focus only on one block and omit the superscript (β).

In practical systems, the channel vector h is estimated atthe receiver as [38]

h = h + he, (8)

where he ∈ Cn×1 represents the vector of channel estimationerrors with the distribution CN (0, σ 2

e In), and it is independentof h. Hence, h, whose elements are hk, k = 1, . . . , n, followsthe distribution CN (0, (1+σ 2

e )In). According to the first-orderautoregressive model and (8), h can be given by [39]

h = ρh + u, (9)

where ρ = 1/(1 + σ 2e ) is the correlation coefficient between

the elements of h and h, and u = [u1, . . . , un]T = (1 −ρ)h − ρhe, which is independent of h with the distributionCN (0, (1 − ρ)In).

Fig. 2. MRC detector for IM-OFDM-SS.

1) ML Detector: From (5), the mismatched ML detectormakes a joint decision on the index of spreading code and theconstellation symbol by searching all possible combinationsof them, namely

(i, s) = arg mini,s

∥∥y − Xh∥∥2

. (10)

Then, the corresponding p information bits can be readilyrecovered from i and s. Obviously, the ML detector shouldsearch all Mn possible X to find out the one that minimizesthe ML metric, which results in the computational complexityin terms of complex multiplications of order ∼ O(Mn) perblock. Although the complexity of ML detector is acceptablefor moderate values of n and M , it still poses intolerable com-putational burden in the case of multi-user communications,as we will see in Section IV. Therefore, the design of a detectorwith complexity lower than that of ML detector is demanding.

2) MRC Detector: In this detector, the detection processis divided into two steps, as described in Fig. 2. The firststep is to estimate the selected spreading code, and the secondstep is to detect the information symbol based on the spreadingcode estimated in the first step.

At the first stage, y is fed into n one-tap equalizers inparallel, each of which is simply realized by one complex-valued multiplication per subcarrier. According to MRC, thel-th equalizer for the k-th subcarrier is given by zl,k = h∗

kc∗l,k ,

and combining n subcarriers, yields

�l =n∑

k=1

zl,k yk, l = 1, . . . , n. (11)

Then, the squared values of the outputs of n branches arecompared to determine the index of the selected spreadingcode via

i = arg maxl

|�l |2. (12)


After obtaining i , at the second stage, the information symbol scan be estimated readily via

s = arg mins

∣∣∣∣∣n∑

k=1

zi,k yk − sn∑

k=1

∣∣hk∣∣2∣∣∣∣∣2

. (13)

From the description above, complex multiplications oforder ∼ O(n) and O(M) are involved at the first and sec-ond stages, respectively. Considering the two-stage detectionprocess is performed successively, we conclude that the overallcomputational complexity of the MRC detector is of order∼ O(n + M), which is very beneficial in multi-user commu-nications. In addition, since MRC is able to obtain a largeminimum distance among |�l |2, l = l, . . . , n, the spreadingcode index detection of the MRC detector has a comparablereliability to that of the ML detector. Once the spreadingcode index is detected correctly, the second step of the MRCdetector becomes the ML detection. Therefore, the MRCdetector has the ability to achieve the near-ML performance,which will be validated in Section V.

Remark 1: Channel estimation errors can be regardedas interference, which lowers the signal-to-interference-plus-noise ratio (SINR). Moreover, with increasing SNR, the SINRconverges towards a constant. Therefore, for the ML detection,channel estimation errors deteriorate seriously the performanceof the joint detection of the spreading code index and theM-ary symbol. Particularly, an error floor is observed in thehigh SNR region. For the MRC detection, channel estimationerrors narrow the minimum distance among |�l |2 at the firststage, which results in a higher error probability of detectingthe spreading code index. The second stage detection alsobecomes worse as a result of a lower SINR. Similar to theML detector, the MRC detector creates an error floor athigh SNR. In the presence of channel estimation errors, sincethe spreading code index is estimated first by comparing npossible energy-like metrics, i.e., |�l |2, the MRC detectoris less susceptible to the channel estimation errors, than theML detector. Additionally, the MRC detector exhibits near-ML performance at much lower complexity. We conclude thatthe MRC detector is more suitable for systems with channelestimation errors.

III. PERFORMANCE ANALYSES

In this section, we derive an upper bound and an analyticalexpression for the BEP of IM-OFDM-SS in the presence ofchannel estimation errors, assuming ML and MRC detection,respectively.

A. BEP Upper Bound for the ML Detector

In the presence of channel estimation errors, the conditionalpairwise error probability (PEP) can be calculated as [9]

Pr(

X → X∣∣h)

= Q

⎛⎜⎜⎝

∥∥∥(

X − X)

h∥∥∥

2

√2σ 2

e

∥∥∥XH(

X − X)

h∥∥∥

2 + 2N0

∥∥∥(

X − X)

h∥∥∥

2

⎞⎟⎟⎠,

(14)

where X is the estimate of X. Since |ci,k |2 = 1 for any i,k ∈ {1, . . . , n}, we have

∥∥∥XH (X − X)h∥∥∥

2 =n∑

k=1

∣∣sci,k∣∣2∣∣(xk − xk

)hk∣∣2

= 1

n‖x‖2

∥∥∥(X − X)h∥∥∥

2. (15)

Hence, (14) can be expressed as

Pr(

X → X∣∣h)

= Q

⎛⎜⎜⎜⎝

√√√√√∥∥∥(

X − X)

h∥∥∥

2

2N0 + 2σ 2e ‖x‖2/n

⎞⎟⎟⎟⎠. (16)

By following the method presented in [9], we obtain anapproximate unconditional PEP

Pr(

X→X)

≈ 1/12

det(In + q1KnA

) + 1/4

det(In + q2KnA

) , (17)

where q1 = n/(4nN0 +4‖x‖2σ 2e ), q2 = n/(3nN0 +3‖x‖2σ 2

e ),and A = (X − X)H (X − X). Then, according to the unionbounding technique, a BEP upper bound can be given by

Pu ≤ 1

p2p

∑X

∑

X

Pr(

X → X)

N(

X, X), (18)

where N(X, X) is the number of erroneous bits when X isdetected as X.

At high SNR, (18) approximates to

Pu ≈ 1

p2p

∑

X,X

(d∏

ω=1

λω

(KnA

))−1

×(

1

12qd1

+ 1

4qd2

)N(

X, X)

, (19)

where d = rank(KnA) = rank(A) and λω(KnA),ω = 1, . . . , d , are the nonzero eigenvalues of KnA.

Remark 2: The diversity order achieved by IM-OFDM-SS without channel estimation errors is given bydmin = min rank (A). Recalling that A = (X− X)H (X− X) =diag(|sci − sci |2), when the receiver correctly detects thespreading code index and makes a decision error for theM-ary symbol, we have A = diag(|sci − sci |2) that has nozero diagonal element, by which we are led to dmin = n. Whenthe transmitted spreading code is detected erroneously, forZadoff-Chu codes with n > 2 or Walsh codes, dmin is achievedin the case of the correct estimation of the M-ary symbol.For this type of error event, we have A = |s|2diag(|ci − ci |2).Hence, dmin is the minimum Hamming distance between anytwo different orthogonal codes drawn from C, which can beshown to be n/2. On the other hand, for Zadoff-Chu codeswith n = 2, the Hamming distance between two codes is 2.Hence, for M-PAM, we have dmin = 2, while for M-QAM,since there exist combinations of s and s that generate d = 1,we have dmin = 1. To sum up, the achievable diversity order


of IM-OFDM-SS is given by

dmin ={

2, for n = 2, M-PAM, and Zadoff-Chu codes,

n/2, otherwise.

(20)

Compared with classical OFDM and OFDM-IM, a higherdiversity order is obtained by IM-OFDM-SS when n ≥ 4.Besides, IM-OFDM-SS provides a flexible trade-off betweenthe SE and the diversity order by adjusting n.

B. BEP Analysis for the MRC Detector

Motivated by the two-stage nature of the MRC detector,we observe that the bit errors of IM-OFDM-SS result fromtwo error cases:

(i) an incorrect index estimation of the spreading code;(ii) a correct index estimation of the spreading code.

In case (i), due to the incorrect detection of the transmittedspreading code, bit errors occur in both p1 index and p2ordinary modulation bits. In case (ii), although the index of thetransmitted spreading code is correctly detected, there is stilla probability of error in p2 ordinary modulation bits. Sincecases (i) and (ii) are complementary, the overall BEP can beevaluated as

Pb = Pcd Pb,i + (1 − Pcd ) Pb,ii, (21)

where Pb,i and Pb,ii are the BEPs for cases (i) and (ii),respectively, and Pcd is the erroneous spreading code detectionprobability.

In case (i), the index of the spreading code may be detectedincorrectly as one of the remaining n − 1 indices with equalprobability Pcd/(n −1). A different incorrect index may resultin a different number of erroneous bits, and it can be readilyfigured out that there are in total C(p1, t) error events thatcorrespond to t erroneous index bits, where t ∈ {1, . . . , p1}.On the other hand, consider that at high SNR, demodulating anM-PAM/QAM Gray encoded symbol based on an incorrectlydetected spreading code mostly yields one erroneous bit [8].Therefore, we have

Pb,i ≈ 1

p

(1

n − 1

p1∑t=1

tC (p1, t) + 1

). (22)

In case (ii), Pb,ii can be easily formulated as

Pb,ii = p2

pPsb, (23)

where Psb is the BEP of M-ary PAM/QAM demodulation overn i.i.d. Rayleigh fading channels.

From (21)-(23), the overall BEP for the MRC detectorof IM-OFDM-SS can be derived once the Pcd and Psb

are obtained. Next, we will deal with the calculation ofPcd and Psb.

1) Erroneous Spreading Code Detection Probability (Pcd ):At the first stage, the receiver selects the index with the max-imum value of |�l |2 as the index of the transmitted spreadingcode. Since each spreading code is selected equiprobably atthe transmitter, without loss of generality, we assume ci is

transmitted. Then, the error probability conditioned on thesymbol s and channel vector h is given by

Pcd∣∣s,h = 1 − P

(|�i |2 > |�1|2, . . . , |�i |2 > |�n|2

)

≈ 1 −n∏

l=1l �=i

P (Dil > 0)

= 1 −n∏

l=1l �=i

[1 − P (Dil ≤ 0)], (24)

where Dil = |�i |2 − |�l |2. With (9), the received signal yk

in (6) can be rewritten as

yk = ρshkci,k + sci,kuk + wk, k = 1, . . . , n, (25)

which leads to

�i = ρsn∑

k=1

∣∣hk∣∣2 + s

n∑k=1

h∗kuk +

n∑k=1

h∗kc∗

i,kwk (26)

and

�l = ρsn∑

k=1

∣∣hk∣∣2c∗

l,kci,k + sn∑

k=1

h∗kc∗

l,kci,k uk +n∑

k=1

h∗kc∗

l,kwk,

(27)

for i �= l. It is easy to observe that �i and �l are two complexGaussian random variables with mean values

μi = ρsn∑

k=1

∣∣hk∣∣2 and μl = ρs

n∑k=1

∣∣hk∣∣2c∗

l,kci,k , (28)

respectively. The variances of the real and imaginary compo-nents of �i and �l are the same, given by

σ 2ii = σ 2

ll = 1

2

(|s|2 (1 − ρ) + N0

) n∑k=1

∣∣hk∣∣2. (29)

The cross-covariance between �i and �l is defined as

σ 2il = 1

2E{(�i − μi ) (�l − μl)

∗}

= 1

2

(|s|2 (1 − ρ) + N0

) n∑k=1

∣∣hk∣∣2cl,kc∗

i,k . (30)

Recall that the decision metric Dil in (24) is a specialquadratic form of complex Gaussian random variables.According to [40, Eq. (9A.10)], we have

P (Dil ≤ 0) = Q1 (a, b) − 1

2exp

(−a2 + b2

2

)I0 (ab) , (31)

where

a =[ν (ξ1ν − ξ2)

2

]1/2

, b =[ν (ξ1ν + ξ2)

2

]1/2

, (32)

with

ν =√√√√ 1

4(σ 2

ii σ2ll − ∣∣σ 2

il

∣∣2) ,

ξ1 = 2(|μi |2σ 2

ll + |μl |2σ 2ii − μ∗

i μlσ2il − μiμ

∗l

(σ 2

il

)∗),

ξ2 = |μi |2 − |μl |2. (33)


Fig. 3. 8-PAM constellation.

After obtaining P(Dil ≤ 0) for all l = 1, . . . , n and l �= i ,Pcd

∣∣s,h can be calculated by (24). Averaging Pcd∣∣s,h over s

and h, the unconditional erroneous spreading code detectionprobability can be expressed as

Pcd = Es,h

{Pcd

∣∣s,h}

= 1

M

∑s

∫

hPcd

∣∣s,h f(h)dh. (34)

Unfortunately, due to the complexity of Pcd∣∣s,h , a closed-form

expression for Pcd is not available. However, the calculationof Pcd can be easily performed in computationally efficientsoftwares such as MATLAB using numerical evaluation.

2) BEP of M-ary PAM/QAM Demodulation (Psb): Here,we derive closed-form BEP expressions for M-PAM andM-QAM over n i.i.d. Rayleigh fading channels when then-branch MRC receiver is used. With error-free detection ofthe spreading code, (25) can rewritten as

yk = ρshk + suk + wk︸︷︷︸Equivalent noise

, k = 1, . . . , n, (35)

which shows that the equivalent noise depends on the trans-mitted symbol.

Let us first consider the M-PAM constellation and assumethat M-PAM symbols are transmitted over the I- channel.Fig. 3 illustrates the 8-PAM constellation with the distancevector dI = [d I

1 , d I2 , . . . , d I

pI2]T = [2pI

2−1, 2pI2−2, . . . , 1]T d I

pI2,

where pI2 = 3 and d I

pI2=1. Let us label the constellation points

from the left to right with integers counting from 0 to M − 1,such that the Gray code ( f I

1, j , f I2, j , . . . , f I

pI2 , j

) for the j -th

( j = 0, . . . , M − 1) point in the constellation is given by

f I1, j = bI

1, j ,

f It, j = bI

t, j + bIt−1, j mod 2, t = 2, . . . , pI

2 , (36)

where (bI1, j , bI

b, j , . . . , bIpI

2 , j) is the binary representation of j .

Some other parameters, such as the average symbol energy,the positions of the symbols in the constellation, and the

decision boundaries for each bit, can also be obtained fromthe constellation [41]:

• The average symbol energy is given by Es = (dI)T

dI .• The positions of M symbols in the constellation form the

vector dI ′s = [d I ′

s,0, . . . , d I ′s,M−1]T , where

d I ′s, j =

pI2∑

t=1

(2bI

t, j − 1)

d It , j = 0, . . . , M − 1, (37)

and the constellation with unit average power can beobtained by dI

s = [d Is,0, . . . , d I

s,M−1]T = dI ′s /

√Es .

• The l-th decision boundary of the t-th bit is definedas

B It,l =

d I

s,(2l−1)2pI2−t

+ d I

s,(2l−1)2pI2−t+1

2,

l = 1, . . . , 2t−1. (38)

For the first bit of j -th symbol, i.e., f I1, j , considering (35)

and the decision boundary for f I1, j at the y-axis, an error

will take place if the equivalent noise exceeds the absolutevalue of ρd I

s, j when f I1, j is 0. Therefore, the error probability,

Psb( f I1, j ), for bit f I

1, j is of the form

Psb

(f I1, j

)= F

(−ρd I

s, j

), (39)

where F(x) is given by (40), shown at the bottom of this page,with

μc (x) =√

γ x2

1 + γ x2 , γ = 1/[ρ((1 − ρ)(d I

s, j )2 + N0

)].

(41)

The proof of (39) is provided in Appendix. When f I1, j is 1,

the error probability Psb( f I1, j ) becomes

Psb

(f I1, j

)= F

(ρd I

s, j

)= 1 − F

(−ρd I

s, j

). (42)

Putting (39) and (42) together, we obtain a generic expressionfor Psb( f I

1, j ), namely

Psb

(f I1, j

)= f I

1, j + (−1)f I1, j F

(B I

1,1 − ρd Is, j

). (43)

The analysis given above for f I1, j can be easily generalized

to that for f It, j

(t = 2, . . . , pI

2

)by taking into account the

effects of multiple decision boundaries where the sign of F(·)should be changed one time whenever a new boundary isdetected, resulting in

Psb

(f It, j

)= f I

t, j + (−1)f It, j

2t−1∑l=1

(−1)l+1F(

B It,l −ρd I

s, j

). (44)

F (x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(1 − μc (x)

2

)n n−1∑k=0

C (n − 1 + k, k)

(1 + μc (x)

2

)k

, x ≥ 0

1 −(

1 − μc (x)

2

)n n−1∑k=0

C (n − 1 + k, k)

(1 + μc (x)

2

)k

, x < 0

(40)


By averaging (44) over those symbols in the left of theplane, we obtain the average error probability for the t-th bit(t = 1, . . . , pI

2) as

Psb

(f It

)= 2

M

M/2−1∑j=0

[f It, j + (−1)

f It, j

×2t−1∑l=1

(−1)l+1 F(

B It,l − ρd I

s, j

)]. (45)

Finally, because of the equal transmit probability for eachbit, a closed-form BEP expression for M-PAM over n i.i.d.Rayleigh fading channels can be derived as

Psb = 2

Mlog2 (M)

log2(M)∑t=1

M/2−1∑j=0

[f It, j + (−1)

f It, j ×

2t−1∑l=1

(−1)l+1

F(

B It,l − ρd I

s, j

) ]. (46)

Next, we consider the case for M-QAM constellation withM = 2pI

2+pQ2 , where pI

2 and pQ2 bits are transmitted over the

I- and Q- channels, respectively. Since the M-QAM constella-tion can be regarded as two PAM constellations in quadraturewith one being 2pI

2 -PAM in the I- channel and the other being2pQ

2 -PAM in the Q- channel, the BEP analyses for M-QAMare similar to those for the M-PAM addressed above.

Before the presentation of the BEP expression for M-QAM,we redefine some parameters for the Q- channel includingthe Gray code, distance vector, position vector, and decisionboundaries, in a similar fashion. Note that for M-QAM theaverage symbol energy becomes

Es =(

dI)T

dI +(

dQ)T

dQ, (47)

and the γ in (41) should be revised as γ = 1/ρ[(1 −ρ)((d I

s, j )2 + 1/2) + N0], where for the instantaneous power

of the symbol over Q- channel, we used the average powervalue, i.e., 1/2 for simplicity.

Therefore, the BEP for M-QAM over n i.i.d. Rayleighfading channels is derived as (48) [42], shown at the bottomof this page.

After obtaining Pcd and Psb, the overall BEP for the MRCdetector of IM-OFDM-SS can be derived by substituting (22)and (23) into (21).

IV. EXTENSION TO MULTI-CODE AND

MULTI-USER SCENARIOS

In this section, we extend the idea of IM-OFDM-SS tomulti-code and multi-user scenarios.

Fig. 4. The IM operation of the GIM-OFDM-SS transmitter for theβ-th block.

A. Extension to Multi-Code Scenario

Aiming at increasing the SE of IM-OFDM-SS, in thissubsection, we propose the GIM-OFDM-SS scheme, in whichmultiple spreading codes are activated for one user. Thetransmitter structure of GIM-OFDM-SS is the same as thatof IM-OFDM-SS except that the IM process is replaced bythe multi-code version. We highlight the IM operation for theβ-th block in Fig. 4.

In GIM-OFDM-SS, for the β-th block, na spreadingcodes, {ci(β,1) , . . . , ci(β,na )}, are activated according to thep1 = ⌊

log2(C(n, na))⌋

index bits via the combinatorialmethod [9], while p2 = na log2(M) symbol bits determinena M-ary PAM/QAM symbols, {s(β,1), . . . , s(β,na)}. Then,s(β,τ) is spread by ci(β,τ ) , τ = 1, . . . , na . The summation ofthe spread symbols is given by x(β) = ∑na

τ=1 s(β,τ)ci(β,τ ) .Afterwards, the remaining procedures are the same as thatof IM-OFDM-SS. Obviously, the SE of GIM-OFDM-SSincreases to (bps/Hz)

EGIM−OFDM−SS =⌊

log2 (C (n, na))⌋+ na log2 (M)

n. (49)

At the receiver, both the indices of na spreading codes andna symbols should be estimated for each block. Likewise,the ML and MRC detectors can be derived accordingly bymodifying (10)-(13).

B. Extension to Multi-User Scenario

In this subsection, the idea of IM-OFDM-SS is extendedto multi-user communications, giving rise to the scheme ofIM-MC-CDMA. Fig. 5 shows the block diagram of IM-MC-CDMA, where T users and the base station are all equippedwith the transceivers of IM-OFDM-SS. To distinguish eachuser and enable IM at the same time, the spreading code setC is partitioned into T subsets each of which contains n/Telements, and each user selects the spreading code from thecorresponding subset according to its index bits. Therefore,

Psb = 1

log2 (M)

{21−pI

2

pI2∑

t=1

2pI2−1−1∑j=0

[f It, j + (−1)

f It, j

2t−1∑l=1

(−1)l+1 F(

B It,l − ρd I

s, j

)]

+ 21−pQ2

pQ2∑

t=1

2pQ2 −1−1∑j=0

[f Qt, j + (−1)

f Qt, j

2t−1∑l=1

(−1)l+1 F(

B Qt,l − ρd Q

s, j

)]}. (48)


Fig. 5. Block diagram of IM-MC-CDMA for (a) downlink and (b) uplink.

the overall SE of IM-MC-CDMA is given by (bps/Hz)

EIM−MC−CDMA = T

nlog2 (Mn/T ) . (50)

In the downlink, without loss of generality, we focus on thedetection problem at the first user, whose received signal is

yk = hk

T∑t=1

s(t)ci(t),k + w(1)k , k = 1, . . . , n, (51)

where s(t), ci(t),k , and w(t)k are the transmitted symbol, selected

spreading code, and noise, respectively, on the k-th subcarrier.From (51), both the optimal ML detection and single-userdetection can be employed to detect the data.

The ML detector estimates the data of all T users jointlyvia{

s(1), i (1), . . . , s(T ), i (T )}

= arg min{s(1),i(1),...,s(T ),i(T )}

n∑k=1

∣∣∣∣yk − hk

T∑t=1

s(t)ci(t),k

∣∣∣∣2

. (52)

The computational complexity of (52) is of order∼O((Mn/T )T ) per block in terms of complex multiplications,which grows exponentially with T .

To achieve low complexity detection, we propose a single-user detector based on the MRC. For this detector, the l-thequalizer of the first user for the k-th subcarrier is given by

zl(1),k =h∗

kc∗l(1),k∣∣hk

∣∣2 (T − 1) + N0

(53)

for l = 1, 2, . . . , n/T . Similar to (11) and (12), the estimateof the index of the spreading code for the first user can beobtained through

i (1) = arg maxl(1)

∣∣�l(1)

∣∣2, (54)

where �l(1) =n∑

k=1zl(1),k yk . Correspondingly, we have

s = arg mins

∣∣∣∣∣n∑

k=1

zi ,k yk − sn∑

k=1

∣∣hk∣∣2

∣∣hk∣∣2 (T − 1) + N0

∣∣∣∣∣2

. (55)

As seen from (54) and (55), the computational complexity peruser is of order O(M + n/T ).

In the uplink, we assume perfect synchronization or quasi-synchronization, in which the timing offsets between users aremuch smaller than the symbol duration. In such a scenario,the CP inserted for intersymbol interference prevention canalso be used to contain timing offsets [30]. Moreover, the tim-ing offset in the time domain is naturally transformed to thephase offset in the frequency domain at the receiver, whichcan be incorporated into the subcarrier channel estimation.Therefore, with perfect channel estimation the received signalat the base station is given by

yk =T∑

t=1

ht,ks(t)ci(t),k + wk, k = 1, . . . , n, (56)

where ht,k is the coefficient of the subchannel assigned touser t . Similar to the downlink of IM-MC-CDMA, boththe single-user and multi-user detectors can be employed atthe base station accordingly, which is omitted due to pagelimitations.

V. SIMULATION RESULTS AND COMPARISONS

In this section, the uncoded BER performance of IM/GIM-OFDM-SS and IM-MC-CDMA over Rayleigh fadingchannels is evaluated through Monte Carlo simulations.The BEP upper bounds and analytical BEP expressions arealso examined. To show the superiority of the proposedIM-OFDM-SS scheme, it is compared with classical OFDM,OFDM-IM [9], DM-OFDM [22], CI-OFDM-IM [24], andOFDM-SS [28], while for multi-user systems, the proposedIM-MC-CDMA scheme is compared with the conventionalMC-CDMA [28]. It is worth noting that the primary and thesecondary constellations used by DM-OFDM are designedthrough constellation rotation. For simplicity, we will referto “(CI-)OFDM-IM (n, n′)” as the (CI-)OFDM-IM schemein which n′ out of n subcarriers are active, and “DM-OFDM(n, n′)” as the DM-OFDM scheme in which n′ out of nsubcarriers are modulated by the primary constellation whilethe remaining subcarriers employ the secondary constellation.

A. IM-OFDM-SS Performance

In this subsection, the error performance of theIM-OFDM-SS scheme is assessed. To see the effects ofdifferent spreading codes, in Fig. 6, we present the BERcurves of IM-OFDM-SS schemes with Zadoff-Chu and Walshcodes, where ML detectors, n = 4, and binary PSK (BPSK)are employed with σ 2

e = 0/0.01/0.05. As seen from Fig. 6,the IM-OFDM-SS schemes with Zadoff-Chu codes outperformthose with Walsh codes for σ 2

e = 0/0.01/0.05 throughoutthe considered SNR region, which can be explained by thefact that IM-OFDM-SS schemes with Zadoff-Chu codes havelarger minimum Euclidean distances between transmissionvectors. In particular, we observe that without channelestimation errors, about 3 dB SNR gain at a BER value of10−5 can be achieved by the scheme with Zadoff-Chu codescompared with the scheme using Walsh codes; however, both


Fig. 6. Performance comparison between Zadoff-Chu and Walsh codes ofIM-OFDM-SS with ML detectors (n=4, BPSK, σ2

e = 0/0.01/0.05).

Fig. 7. Performance comparison between ML and MRC detectors ofIM-OFDM-SS (n = 4, 4/16-QAM, σ 2

e = 0/0.01).

of them have the same diversity order of two. The channelestimation errors deteriorate the performance greatly anderror floors are observed in the high SNR region due to themismatched ML detectors. However, the error floors withZadoff-Chu codes are lower than those with Walsh codes.In addition, Zadoff-Chu codes result in the lower peak toaverage power ratio [28]. Due to these advantages, Zadoff-Chucodes are adopted in the rest of computer simulations.

Fig. 7 presents the performance comparisons between MLand MRC detectors of IM-OFDM-SS, where n = 4 and4/16-QAM are employed with σ 2

e = 0/0.01. To verify theanalyses given in Section III, in the figure, we also plot theBEP upper bounds (18) and the theoretical BEP curves (21)for ML and MRC detectors, respectively. For the clarity of

Fig. 8. Performance comparison among IM-OFDM-SS, GIM-OFDM-SS,classical OFDM, OFDM-IM, CI-OFDM-IM, and OFDM-SS at an SEof 1 bps/Hz, where ML detectors are used for all schemes.

Fig. 9. Performance comparison among IM-OFDM-SS, classical OFDM,OFDM-IM, DM-OFDM, CI-OFDM-IM, and OFDM-SS at an SE levelof 1.5 bps/Hz, where ML detectors are used for all schemes.

presentation, the BEP upper bounds without channel estima-tion errors are provided in Figs. 8 and 9. It can be observedfrom Fig. 7 that MRC detectors exhibit near-ML performancewith/without channel estimation errors. For σ 2

e = 0, bothdetectors achieve the same diversity order and less than 1 dBSNR loss is incurred by the MRC detectors compared withthe ML detectors. Besides, the BEP upper bounds becometight in the high SNR region and the approximate theoreticalBEP curves agree with their simulation counterparts very well,no matter whether there exist channel estimation errors or not.

In Figs. 8–10, the BER performance of ML detection ofIM-OFDM-SS without channel estimation errors is evaluatedat various SEs. In Fig. 8, we compare the BER performance of


Fig. 10. Performance comparison among IM-OFDM-SS, classical OFDM,OFDM-IM, DM-OFDM, CI-OFDM-IM, OFDM-SS, and SM at an SE levelof 1.5–2 bps/Hz, where ML detectors are used for all schemes.

IM-OFDM-SS with n = 4 (8) and 4-QAM (32-QAM), GIM-OFDM-SS with n = 16, na = 5, and BPSK, classical OFDMwith BPSK, OFDM-IM with (n, n′) = (4, 2) and BPSK, CI-OFDM-IM with (n, n′) = (4, 2) and BPSK, and OFDM-SSwith n = 4 and 16-QAM at an SE of 1 bps/Hz. All schemesemploy ML detectors. From Fig. 8, we observe that the the-oretical BEP upper bounds of IM-OFDM-SS agree with theircomputer simulation counterparts very well in the high SNRregion. GIM-OFDM-SS, IM-OFDM-SS with n = 8, OFDM-SS, IM-OFDM-SS with n = 4, CI-OFDM-IM, OFDM-IM,and classical OFDM achieve diversity orders of 8, 4, 4, 2, 2,1, and 1, respectively. The diversity order of IM/GIM-OFDM-SS increases with n and therefore, GIM-OFDM-SS with thehighest diversity order performs the best in the SNR regionof interest. Both IM-OFDM-SS schemes perform better thanclassical OFDM, OFDM-IM, and CI-OFDM-IM for all SNRvalues. More specifically, in spite of the same diversity order,IM-OFDM-SS with n = 4 and 4-QAM obtains approximately3 dB SNR gain at a BER value of 10−5 over CI-OFDM-IM.Furthermore, IM-OFDM-SS with n = 8 and 32-QAM outper-forms OFDM-SS throughout the considered SNR region andperforms better than IM-OFDM-SS with n = 4 and 4-QAM athigh SNR. Contrarily, because of the larger constellation size,IM-OFDM-SS with n = 8 and 32-QAM exhibits a worse BERthan that with n = 4 and 4-QAM in the low SNR region.

Fig. 9 depicts the comparison results between IM-OFDM-SS with n = 4 and 16-QAM, classical OFDM withBPSK/4-QAM, OFDM-IM with (n, n′) = (4, 2) and 4-QAM,DM-OFDM with (n, n′) = (4, 2) and BPSK, CI-OFDM-IMwith (n, n′) = (4, 2) and 4-QAM, and OFDM-SS with n = 4and 64-QAM at an SE level of 1.5 bps/Hz. In DM-OFDM,two out of four subcarriers are modulated by the conventionalBPSK, while the others employ π/2-rotated BPSK. MLdetectors are used for all schemes and an SE of 1.5 bps/Hz isachieved except classical OFDM with BPSK (4-QAM) that

Fig. 11. Performance comparison between IM-MC-CDMA and conventionalMC-CDMA in the downlink, where ML and MRC detectors are used.

has an SE of 1 bps/Hz (2 bps/Hz). Although it is not providedin the figure, the curve for classical OFDM with an SEof 1.5 bps/Hz can be predicted to be lying between those at1 bps/Hz and 2 bps/Hz. As seen from the figure, IM-OFDM-SS performs the best among all schemes in the SNR region ofinterest. Although achieving a higher diversity order, OFDM-SS merely exhibits its superiority at high SNR due to its largerconstellation size. DM-OFDM and OFDM-IM achieve a smallSNR gain over classical OFDM; however all of them havethe same diversity order of unity. DM-OFDM and OFDM-IMalmost exhibit the same BER performance, which can beunderstood by the fact that the power saving with a factorof 50% stemming from the inactive subcarriers, correspondingto 10log10(2) ≈ 3 dB SNR gain provided by OFDM-IM,is counterbalanced by the modulation loss from BPSK to4-QAM. On the other hand, IM-OFDM-SS achieves about3 dB SNR gain over CI-OFDM-IM at a BER value of 10−5.

The BER performance of IM-OFDM-SS with n = 4 and64-QAM, classical OFDM with 4-QAM, OFDM-IM with(n, n′) = (4, 3) and 4-QAM, DM-OFDM with (n, n′) = (4, 2)and BPSK, CI-OFDM-IM with (n, n′) = (8, 6) and 4-QAM,OFDM-SS with n = 4 and 256-QAM, and plain SMwith 2 transmit and 1 receive antennas at an SE levelof 1.5–2 bps/Hz is compared in Fig. 10. ML detectors areused for all schemes and an SE of 2 bps/Hz is achievedexcept DM-OFDM, which has an SE of 1.5 bps/Hz. Similarto the observations in Figs. 8 and 9, IM-OFDM-SS andCI-OFDM-IM achieve a higher diversity order than classicalOFDM, OFDM-IM, DM-OFDM, and SM. Approximately3 dB SNR gain can be obtained at a BER value of 10−5 byIM-OFDM-SS compared with CI-OFDM-IM. Furthermore,in spite of a lower diversity order, IM-OFDM-SS performsbetter than OFDM-SS in the considered SNR region.

B. IM-MC-CDMA Performance

Fig. 11 shows the performance comparison betweenIM-MC-CDMA and conventional MC-CDMA in the downlink


TABLE I

COMPARISON OF DETECTION COMPLEXITY AMONG IM-OFDM-SS, CLASSICAL OFDM, OFDM-IM, AND OFDM-SS

TABLE II

COMPARISON OF DETECTION COMPLEXITY BETWEEN IM-MC-CDMA AND MC-CDMA

Fig. 12. Performance comparison between IM-MC-CDMA and conventionalMC-CDMA in the uplink, where ML detectors are used.

at the overall SEs of 1 and 2 bps/Hz, where ML and MRCdetectors are used for each scheme. For a fair comparison,the system parameters of IM-MC-CDMA and conventionalMC-CDMA are set to achieve the same SE with the samenumber of users. It can be seen from Fig. 11 that for bothIM-MC-CDMA and conventional MC-CDMA, the MRCdetectors suffer from performance loss compared with MLdetectors due to the multiple access interference. On theother hand, at both SEs, IM-MC-CDMA performs better thanthat of MC-CDMA in the SNR region of interest either forML or MRC detection.

Fig. 12 gives the BER versus SNR curves forIM-MC-CDMA and conventional MC-CDMA in the uplinkwith ML detector at the base station. The system parameters

are the same as those in Fig. 11. From Fig. 12, we observethat with the ML detector, IM-MC-CDMA outperformsconventional MC-CDMA at the SEs of 1 and 2 bps/Hz.

C. Complexity Comparison

In this subsection, the detection complexities ofIM-OFDM-SS and IM-MC-CDMA are compared with thoseof other competitive schemes. For the single-user scenario,Table I lists the average number of metric calculationsper subcarrier resulting by the ML and MRC detectors ofIM-OFDM-SS, and ML detectors of classical OFDM, OFDM-IM, and OFDM-SS, which are calculated as M , 1 + M/n, M ,Mn′

2�log2(C(n,n′))�/n, and M/n, respectively. For the down-link/uplink multi-user communications, Table II presents theaverage number of metric calculations per subcarrier resultingby ML and MRC detectors of IM-MC-CDMA and those detec-tors of MC-CDMA, which are calculated as (Mn/T )T /n,1 + T M/n, MT /n, and T M/n, respectively. For a faircomparison, we select appropriate parameters for each schemeto obtain the same SEs. As shown in Tables I and II, ourproposed IM-OFDM-SS and IM-MC-CDMA schemes achievecomparable detection complexities to other competitors.

VI. CONCLUSION

In this paper, we have proposed IM-OFDM-SS, in whichthe techniques of SS and IM are combined based on theframework of OFDM to improve the performance of existingOFDM, OFDM-IM, and OFDM-SS schemes. IM-OFDM-SSspreads an M-ary modulated symbol over several subcarriersby a spreading code and utilizes the index of the selectedspreading code to convey additional bits. A low-complexityMRC-based detector has been designed for IM-OFDM-SS.An upper bound and an approximate mathematical expressionfor the BEP have been derived, taking into account the channelestimation errors. Subsequently, two extensions of IM-OFDM-SS have been proposed, including GIM-OFDM-SS in which


Psb

(f I1, j

)= 1

π

∫ π/2

0

⎡⎢⎣∫ ∞

0exp

⎛⎜⎝−

γk

(−ρd I

s, j

)2

sin2φ

⎞⎟⎠ fγk (γk) dγk

⎤⎥⎦

n

dφ = 1

π

∫ π/2

0

⎛⎜⎝ sin2φ

γ(−ρd I

s, j

)2 + sin2φ

⎞⎟⎠

n

dφ

=(

1 − μc

2

)n n−1∑k=0

C (n − 1 + k, k)

(1 + μc

2

)k

(63)

multiple spreading codes are activated for one user to improvethe SE, and IM-MC-CDMA in which each user/base station isequipped with the IM-OFDM-SS transceiver. Finally, the BERperformance of IM/GIM-OFDM-SS and IM-MC-CDMA hasbeen simulated extensively, whose results show that IM/GIM-OFDM-SS significantly outperforms classical OFDM aswell as the existing OFDM-IM and OFDM-SS schemes,and IM-MC-CDMA performs better than conventionalMC-CDMA at the same SEs for either multi-user orsingle-user detection.

We remark that in the presence of asynchronous users,the performance of IM-MC-CDMA would be deterio-rated as in the conventional MC-CDMA. The performanceanalysis for this case will be considered as our futurework.

APPENDIX

PROOF OF (39)

The conditional error probability for bit f I1, j ,

Psb( f I1, j |{γ }n

k=1), is given by

Psb

(f I1, j |{γk}n

k=1

)= Q

(√2γtotal

(−ρd I

s, j

)), (57)

where γtotal = ∑nk=1 γk is the total instantaneous conditional

SNR with

γk =∣∣hk

∣∣2

(1 − ρ)(

d Is, j

)2 + N0

. (58)

To obtain the average error probability, Psb( f I1, j ), (57) has

to be statistically averaged over the joint PDF of {γk}nk=1. Since

random variables {γk}nk=1 are assumed to be independent,

we have

Psb

(f I1, j

)=∫ ∞

0· · ·

∫ ∞

0︸︷︷︸n− f old

Psb( f I1, j |{γk}n

k=1)

×n∏

k=1

fγk (γk)dγ1 · · · dγn, (59)

where fγk (γk) = exp(−γk/γ )/γ is the PDF of γk with γ =1/[ρ((1 − ρ)(d I

s, j )2 + N0)].

If −ρd Is, j ≥ 0, by using the alternative representation of

Gaussian Q-function

Q (x) = 1

π

∫ π/2

0exp

(− x2

2sin2φ

)dφ, x ≥ 0, (60)

(57) can be expressed as

Psb

(f I1, j |{γk}n

k=1

)= 1

π

∫ π/2

0exp

⎛⎜⎝−

γtotal

(−ρd I

s, j

)2

sin2φ

⎞⎟⎠ dφ

= 1

π

∫ π/2

0

n∏k=1

exp

⎛⎜⎝−

γk

(−ρd I

s, j

)2

sin2φ

⎞⎟⎠dφ.

(61)

Substituting (61) into (59) results in

Psb

(f I1, j

)=∫ ∞

0· · ·

∫ ∞

0︸︷︷︸n− f old

1

π

∫ π/2

0

n∏k=1

exp

⎛⎜⎝−

γk

(−ρd I

s, j

)2

sin2φ

⎞⎟⎠

× fγk (γk) dφdγ1 · · · dγn. (62)

Interchanging the order of integration and grouping the termsof index k, we can formulate (62) as (63), shown at the top

of this page, where μc =√

γ (−ρd Is, j )

2

1+γ (−ρd Is, j )

2 and in the derivation

of the last equality, we have considered [40, eq. (5A.4b)].If −ρd I

s, j < 0, using the equation Q(x) = 1 − Q(−x),we obtain

Psb

(f I1, j

)= 1−

(1 − μc

2

)n n−1∑k=0

C(n − 1 + k, k)

(1 + μc

2

)k

.

(64)

Therefore, putting (63) and (64) together yields (39),completing the proof.

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Qiang Li received the B.S. degree from the InnerMongolia University of Science and Technology,Baotou, China, in 2013, and the M.S. degree fromthe Nanjing University of Aeronautics and Astro-nautics, Nanjing, China, in 2016. He is currentlypursuing the Ph.D. degree with the South ChinaUniversity of Technology, Guangzhou, China. Hiscurrent research interests include MIMO systems,index modulation, and OFDM.

Miaowen Wen (M’14) received the B.S. degreefrom Beijing Jiaotong University, Beijing, China,in 2009, and the Ph.D. degree from Peking Univer-sity, Beijing, in 2014. From 2012 to 2013, he was aVisiting Student Research Collaborator with Prince-ton University, Princeton, NJ, USA. He is currentlyan Associate Professor with the South China Uni-versity of Technology, Guangzhou, China. He hasauthored a book and over 80 papers in refereedjournals and conference proceedings. His researchinterests include index modulation, non-orthogonal

multiple access, physical layer security, and molecular communications.He was a recipient of the Excellent Doctoral Dissertation Award from Peking

University and the Best Paper Awards from the IEEE International Con-ference on Intelligent Transportation Systems Telecommunications in 2012,the IEEE International Conference on Intelligent Transportation Systemsin 2014, and the IEEE International Conference on Computing, Networkingand Communications in 2016. He was an Exemplary Reviewer of the IEEECOMMUNICATIONS LETTERS in 2017. He currently serves as an AssociateEditor for the IEEE ACCESS and is on the Editorial Board of the EURASIPJournal on Wireless Communications and Networking, the ETRI Journal, andthe Physical Communication (Elsevier).


Ertugrul Basar (S’09–M’13–SM’16) received theB.S. degree (Hons.) from Istanbul University,Turkey, in 2007, and the M.S. and Ph.D. degreesfrom Istanbul Technical University in 2009 and2013, respectively. He was with the Departmentof Electrical Engineering, Princeton University,Princeton, NJ, USA, from 2011 to 2012. He wasan Assistant Professor with Istanbul Technical Uni-versity from 2014 to 2017, where he is currently anAssociate Professor of electronics and communica-tion engineering. He is an inventor of two pending

patents on index modulation schemes. His research interests include MIMOsystems, index modulation, cooperative communications, OFDM, and visiblelight communications.

Recent recognition of his research includes the Turkish Academy ofSciences Outstanding Young Scientist Award in 2017, the first-ever IEEETurkey Research Encouragement Award in 2017, and the Istanbul TechnicalUniversity Best Ph.D. Thesis Award in 2014. He was also a recipient of fourbest paper awards, including one from the IEEE International Conference onCommunications in 2016. He has served as a TPC member for several IEEEconferences. He is a regular reviewer for various IEEE journals. He currentlyserves as an Associate Editor for the IEEE COMMUNICATIONS LETTERS andthe IEEE ACCESS, and as an Editor of the Physical Communication (Elsevier).

Fangjiong Chen (M’06) received the B.S. degreein electronics and information technology fromZhejiang University, Hangzhou, China, in 1997, andthe Ph.D. degree in communication and informationengineering from the South China University ofTechnology, Guangzhou, China, in 2002. He waswith the School of Electronics and InformationEngineering, South China University of Technology.From 2002 to 2005, he was a Lecturer, and from2005 to 2011, he was an Associate Professor withthe South China University of Technology.

He is currently a full-time Professor with the School of Electronics andInformation Engineering, South China University of Technology. He is theDirector of the Department of Underwater Detection and Imaging, MobileUltrasonic Detection National Research Center of Engineering Technology.His research interests include signal detection and estimation, array signalprocessing, and wireless communication.

Professor Chen received the National Science Fund for Outstanding YoungScientists in 2013. He was elected to the New Century Excellent TalentProgram of MOE, China, in 2012.

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